SOLUTION: please help me with this question. Express in terms of odd and even numbers why the number 286 would not appear in the series 4, 12, 24, 40... ?

Question 500890: please help me with this question.
Express in terms of odd and even numbers why the number 286 would not appear in the series 4, 12, 24, 40... ?

Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i believe the answer is because 286 is not divisible by 4.
all the number you see there are divisible by 4.
the pattern appears to be that the second number is increased by 8 and the third number is increased by 12 and the fourth number is increased by 16.
this indicates the 5th number wiall be increased by 20, the next after that by 24, the next after that by 28, etc.
all numbers will be divisible by 4.
286 is not divisible by 4 and so will not appear in the sequence.
here's the sequence up to that number.
n = the value of n which is the number of the term in the sequence.
number = the value of the number starting with 4
difference 1 is the difference between that number and the number that is right before it. example: when n = 2, difference 1 is equal to 12 - 4. when n = 3, difference 1 is equal to 24 - 12, etc.
difference 2 is the difference between the difference 1. when n = 3, difference 2 is equal to 12 - 8. when n = 4, difference 2 = 16 - 12.
you can see that difference 2 is constant, i.e. the difference between difference 1 for n and n - 1 will always be 4.

n number difference 1 difference 2 1 4 2 12 8 3 24 12 4 4 40 16 4 5 60 20 4 6 84 24 4 7 112 28 4 8 144 32 4 9 180 36 4 10 220 40 4 11 264 44 4 12 312 48 4 13 364 52 4 14 420 56 4 15 480 60 4 16 544 64 4 17 612 68 4 18 684 72 4 19 760 76 4 20 840 80 4

this leads to the formula for this particular series.
that formula is a(n) = 4 * sum(i) for i = 1 to n.
when n = 1, this becomes 4 * sum(i) for i = 1 to 1 which which becomes 4*1 = 4.
when n = 2, this becomes 4 * sum(i) for i = 1 to 2 which means 4*(1+2) = 4*3 = 12
when n = 3, this becomes 4 * sum(i) for i = 1 to 3 which means 4*(1+2+3) = 4*6 = 24
when n = 4, this becomes 4 *(sum(i) for i = 1 to 4 which means 4*(1+2+3+4) = 4*10 = 40
etc.
all of these numbers are divisible by 4.
286 is not.
you can also see that the number right before 286 is 264 and that the number after that is the 12th number in the sequence so the value of that number will be:
4 * sum(i) for i = 1 to 12 which means 4*(1+2+3+4+5+6+7+8+9+10+11+12) which means 4 * 78 which becomes 312.
286 is bypassed completely.
the simple answer is that 286 is not divisible by 4.
every number in the series is divisible by 4.
that should be enough by itself for you to answer the question.
if that's not sufficient, then the formula should show very clearly that 286 could never be a number in this series, nor any of the numbers between 265 and 311.
Answer by ikleyn(52907) (Show Source): You can put this solution on YOUR website!
.

The question in this post is posed incorrectly.

The correct question should ask

Express in terms of divisibility of integer numbers why the number 286 would not appear in the series 4, 12, 24, 40... ?

No one professionally written textbook will present an assignment in this form.

It tells me that you try to compose a Math problem on your own,
without having knowledge on how it should be done in Math.

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